Re: Major problems with Kaplan -Yorke dimension?
From: Dr Chaos (mbkennelSPAMBEGONE_at_NOSPAMyahoo.com)
Date: 09/08/04
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Date: Wed, 8 Sep 2004 03:30:54 +0000 (UTC)
On Mon, 06 Sep 2004 17:20:24 GMT, Roger L. Bagula <rlbtftn@netscape.net> wrote:
> Dear Dr Chaos,
> Did you know there is another older "Dr Chaos"?
> His site has the Italian guy's Mathematica Lyapunov exponent program
> that was published by Mathematica.
>
> I am going to "top answer as your method makes it very difficult to read
> or reply to a post. And the results begin after a while to resemble
> noodles.
>
> On the fractal dimension of the Rationals ( if you had been paying
> attention to my posts) there are two very different answers:
> Hausdorff--> 0
> Box counting-->1
> My indications are that the real answer is:
> 0.1< s <0.5
> And yes, the Kaplan -Yorke method isn't made for this
> ( as far as I know only Moran similarity dimension
> works well for this region and you need a good equation for the set!).
> As to the symmetricalness of the Jacobian I don't know if that is
> necessarily true unless you confine yourself to Riemannian geometry.
> In the general sense of Sl(n,r) symmetry of the matrix isn't specified.
> just the measure of the determinant of the matrix.
> The Matrix:
> L(i,j)=log(Sqrt(dx(i)/dt*dx(j)/dt))
> has it's problems, but not as great as the Jacobian definition that you
> have used ( which isn't a true Jacobian technically if you look at the
> definition! which I did a while back in trying to understand
> what a young friend was talking about).
> Some where in the texts , somebody in chaos theory has been pushing this
> erroneous idea of the Jacobian around. The true Jacobian comes from
> tensor theory and is about translating one set of coordinated to
> another.
The coordinates in question here are in the same space, from
'now' to the future.
> Your definition is a Markov matrix with determinant one that
> takes the vector of derivatives:
> vd={d(x1)/dt) , ... dx(n)/dt)
> to the vector
> v=(x(1),...,x(n)}
> vd=M*v
The matrix I was referring to was for the finite time
evolution operator, as in a map.
What you do in practice is get it for increments of time '\delta t'
along a typical trajectory, and successively multiply them using a
recursive QR in order to preserve numerical stability.
> It isn't a Jacobian matrix as far as I can tell from the way it is
> defined in "most" physics books. I can give you references if you like.
> vd is tangent to v and isn't a linear transform of coordinates, but an
> operational matrix ( one of the reasons they talk about Hamiltonian
> systems a lot here!)
Whatever you want to call it, it is one way how the Lyapunov exponents
are defined/computed.
> Arguing about something some guy like Ruelle of Kaplan screwed up (I'm
> pretty sure it is a Russian origin definition) and has ph.d's spouting
> all over the place isn't going to get us anywhere!
> The point is that the vector vd is the one that gives the limiting
> Lyapunov exponents and they can't be derived axiomatically since it
> isn't a "true" Jacobian. If it were than the derivation is nearly
> transparent in tensor terms.
what do you mean "axiomatically"?
>
> On the Zipf law: it is the basis of the log -log plots used for entropy
> scales in the box counting ( domain counting) capsicity methods of
> finding dimension. It is one of the basic fractal dimension laws that
> refer to experimental physical data. It is thus, the actual basis for
> much of what we call fractal dimension theory. Frequency domains are
> Fourier transform like in many ways or Laplace transform like
> and are very much like the log(dx(i)/dt) in their domain.
> As far as I know no one has connected the dots here in theory, but
> it appears to exist.
OK, if it is, write a paper.
> I don't think dismissing it as you do is the not
> mark of mature reflection of a scientist. Afterall all science is based
> on statistical information which is only modeled by mathematical equations.
> Some people lose sight of that.
OK, there may be some undiscovered connection in certain circumstances,
but it is not something that is well known.
> I did a bad derivation of entropy from Lyapunov exponents a while back
> that wasn't very satisfying, but I think if you take the convex hull
> maximum as a radius / modulus and divide all the domain points by it
> that you can define a -p*log(p) entropy from a v (v=(x(1),...,x(n)}) set
> of domain points.
> vp=(x(1),...,x(n)}/Abs(Max(v))
> Using the starting point as H0 , a dimension of the entropy type can be
> defined as a Shannon information entropy ( which is actually better
> accepted than the Kolomogorov one in physics/ engineering circles).
D_1, the information dimension has a well known similarity to Shannon
formulae.
> Let us be clear about what we are talking about: we are talking about
> generalized chaotic manifold that are contained in a domain of n
> dimensions. Not a coordinate set that gives a definite surface, but one
> that has a domain that is contained in a definite n dimensional
> "surface". Mostly we really only do 2d, 3d, 4d for these.
> I've worked intensely with these for about 20 years now.
> I've done a lot of new work on them in the last two years besides CMC
> minimal surface, soliton, and differential geometry research.
> I didn't get Lyapunov stuff right the first time through.
> I'm trying to do better, but I admit to be still learning.
> Unlike some ph.d.'s who think what they have learned and past tests on
> is the gospel truth and unchanging set in stone, etc.
> Pretty much, if it wasn't an experimental science that developed over
> time we wouldn't have to study it and do experiments, would we?
> There is a lot of overlap between statistical mechanics and
> statistical theormodynamics and fliud mechanics in the chaos theory area.
>
> I'm sure that you are smarter and probably better trained than me,
> but that doesn't make you right here. Kaplan-Yorke theory is seriously
> faulty in some aspects ( you get different answers for the dimension
> with different starting points! )
the Lyapunov exponents are supposed to be invariants and depend
on the measure and derivatives, not the starting point of some orbit.
> It doesn't give invarient answers.
in what sense is it not sufficiently invariant for your desire?
> Yet it is the closest thing we have to a "true" experimental fractal
> dimension from the actual equations.
and the measure and attractor that they induce upon iteration.
> There is a fluid dynamic version
> of dimension, but I haven't seen a lot about it outside of fluid
> mechanics texts. Dr. McMullen has a paper with a lot more
> but most aren't much different than the others and tend to be in three
> classes:tue
> 1) Measures ( Hausdorff type)
> 2) Entropy ( Kolmogorov type)
> 3) Lyapunov ( differential and roughness type)
> I'm just doing my best to understand them.
> Dr Chaos wrote:
>> On Sun, 05 Sep 2004 15:58:30 GMT, Roger Bagula <tftn@earthlink.net> wrote:
>>
>>>Major problems with Kaplan -Yorke dimension?
>>>1) It isn't really derivable axiomatically?
>>>2) The argument assumes that the Lyapunov exponents are "constants" on
>>>the domain
>>>which if you have done much on these systems, you know isn't true.
>>>And actually what I have read suggests that it fails completely for
>>>Bescovitch -Usell/ Mandelbrot cartoon noise/ chaos
>>>and Weierstrass fractal functions because the differential problems with
>>>these functions.
>>
>>
>> the classic Lyapunov exponents presuppose a deterministic dynamics.
>>
>> Those other systems define fractal measures by other means, e.g.
>> stochastic systems.
>>
>> It is not surprising that the Lyapunov dimension is inapplicable.
>>
>>
>>>Intuitively you suspect when g(i.j) for a manifold is an nbyn symmetric
>>>matrix and the Kaplan-Yorke dimension in based on an
>>>n-vector that something is being missed in the analysis?
>>>Suppose instead of a vector you had a matrix :
>>>L(i,j)=log(Sqrt(dx(i)/dt*dx(j)/dt))
>>>to get a vector solution:
>>>Eigenvalues[L(i,j)]=v(i,i)
>>>d=Sum[v(i,i)/Max[v(i,i)],{i,1 to n}]
>>>Here you get the problem that some Eigenvalue roots come out complex!
>>>So there are problems with Kaplan -Yorke dimension,
>>
>>
>> Not for that reason.
>>
>> The right way to do it for ergodic chaos is to get the
>> time-dependent Jacobian matrix J(x,t), starting at location
>> x and going for time t.
>>
>> Then the eigenvalues of
>>
>> [transpose(J)*J] ^{1/2t}
>>
>> tend to the exponential of the Lyapunov exponents as t-> infinity and
>> become independent of x for all points x in the basin of attraction of
>> the attractor in question. The matrix in question is symmetric and
>> hence has real eigenvalues.
>>
>> Or, without the transpose, you ask for the 'principal values',
>> which are also always real.
>>
>>
>>>yet as far as I can tell it is the most "practical" dimension as a
>>>measure that has been developed
>>>with all the short comings.
>>>That's why it is called a conjecture? It can't be "proved' so far.
>>>I've been using a modified version of it in work on estimates of the
>>>fractal dimension of the Rational numbers
>>>without a lot of success.
>>
>>
>> any sensible definition should yield 1.
>>
>>
>>>What relationship does the frequency type Zipf law dimension have to
>>>Lyapunov?
>>
>>
>> None, Zipf law is a statistical observation and mostly in symbolic discrete
>> space as far as I'm aware.
>>
>>
>>>Can an entropy based on Lyapunov exponents be derived?
>>
>>
>> Yes, the sum of the positive Lyapunov exponents is usually the
>> Kolmogorov-Sinai entropy rate.
>>
>> If you have a symbolic generating partition for the dynamics, then the
>> Shannon entropy rate of the symbolic dynamics will equal the
>> Kolmogorov-Sinai entropy rate.
>>
>>
>>>So that the Kaplan-York dimension can be related to the capacity dimension?
>>
>>
>> don't know.
>>
>>
>>>Respectfully, Roger L. Bagula
>>
>
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